On certain classes of harmonic P-valent functions by applying the Ruscheweyh derivatives

Abstract

In this paper we have introduced two new classes HRp(?, ?, k, v),HRp(?, ?, k, v) of complex valued harmonic multivalent functions of the form f = h+g, where h and g are analytic in the unit disk ? = {z: |z| < 1} and f(z) satisfying the condition Re (1-?)Df+ ?(1-k)(Df)'+ ?k(Df)''> ? A sufficient coefficient condition for this function in the class HRp(?, ?, k, v) and a necessary and sufficient coefficient condition for the function f in the class HRp(?, ?, k, v) are determined. We investigate inclusion relations, distortion theorem, extreme points, convex combination and other interesting properties for these families. .